The effects of wing inertial forces and mean stroke angle on the pitch dynamics of hovering insects

This paper discusses the wing inertial effects and the important role of the mean stroke angle on the pitch dynamics of hovering insects. The paper also presents a dynamic model appropriate for averaging and discusses the pitch stability results derived from the model. The model is used to predict the body angle of five insect species during hover, which are in good agreement with the available experimental results from different literature. The results suggest that the wing inertial forces have a considerable effect on pitch dynamics of insect flight and should not be ignored in dynamic analysis of hovering insects. The results also suggest that the body of hovering insects can not be vibrationally stabilized in a non-vertical orientation. Instead, the pitch angle of a hovering insect’s body is mainly due to a balance of the moment of the insect’s weight and the aerodynamic moment due to flapping kinematics with a nonzero mean stroke angle. Experiments with a flapping wing device confirm this results. To clearly explain the used model and clarify the difference between vibrational and non-vibrational stabilization, first this paper discusses the vibrational control of a three-degree-of-freedom force-input pendulum with its pivot moving in a vertical plane.


Since
Stephenson's observation that an inverted pendulum can be stabilized in its upright orientation by fast vertical vibrations of its pivot in the early twentieth century 1 , and the theoretical explanation of that phenomenon by Kapitza in the mid-twentieth century 2 , the Stephenson-Kapitza pendulum, usually called the Kapitza pendulum, has been the classical example of vibrational control and vibrational mechanics 3,4 .The dynamics, stability analysis, and the mechanics underlying the stability of the Kapitza pendulum are discussed in different literature, for example [5][6][7][8][9][10] .
Developed by Meerkov, vibrational control is changing the stability properties of a dynamical system by introducing high-frequency, zero-mean inputs to the system 3,11 .The Kapitza pendulum benefits from the stabilizing effects of high-frequency, zero-mean, periodic inputs on mechanical systems for its stability.A well-developed, useful method for the dynamic analysis of mechanical systems with high-frequency periodic inputs is averaging.Using the averaging techniques, a time-periodic dynamical system can be approximated by a time-invariant system, called the averaged dynamics.For "high enough" frequencies, the existence of an asymptotically stable equilibrium point of the averaged dynamics guarantees the existence of an asymptotically stable periodic orbit of the time-periodic system in a small neighborhood of that equilibrium point 12,13 .Using the chronological calculus developed in 14 and a series expansion that describes the evolution of mechanical systems subject to timevarying inputs 15 , Bullo developed a closed form for the averaged dynamics of a class of control-affine mechanical systems 16,17 .The inclined Kapitza pendulum discussed in this paper belongs to this class of systems.Therefore, this paper uses the mentioned closed form averaging formula for stability analysis and vibrational control of the force-input inclined Kapitza pendulum and for dynamic analysis of the pitch motion of hovering insects also.
Insect flight has been the inspiration for the development of biomimetic flapping wing vehicles.Design of applicable and efficient flapping wing devices is directly related to our understanding of different aspects of insect flight.The longitudinal motion and pitch stability of hovering insects and flapping wing micro-air vehicles (FWMAVs) are vastly studied in different literature, with the analyses sometimes being inconclusive or the results being in contrast to previous ones.Usually, the mass of the wings, wings inertial forces, the distance between the body center of mass and wing joints (wing hinge or root), and the asymmetry of the stroke (flapping) angle during one flapping cycle are neglected in the analyses [18][19][20][21][22][23][24][25][26][27][28][29][30] .Though in some research the wing inertial forces are also considered in the dynamics, the emphasis in those efforts are on the effects of the considered aerodynamic model and the approximation methods used on the pitch stability and control of insect or FWMAV flight, which, considering the role of aerodynamic forces in flight, is completely justified [31][32][33][34][35][36][37][38][39] .

The 3-DOF force-input pendulum
Consider the 3-DOF pendulum (rigid body) of mass m and mass moment of inertia Ī about its center of mass G depicted in Fig. 1.The orientation angle θ of the pendulum is measured from its down position, that is, the −z -axis.The pivot A of the pendulum, located at a distance d from G, moves in the vertical x-z plane under action of the input force F applied in a fixed direction of angle β with the horizontal x-axis.A constant force, F g = mg , is applied to the pivot in the vertical direction, counteracting the weight of the pendulum.Therefore, the pendulum does not experience any weight, although the moment of the weight about the pivot is still present.An input couple M also acts on the pendulum, as shown in Fig. 1.Small linear dampings with damping coefficients c and c t resist the translational motion and rotation of the body, respectively.
The equations of motion of the system are where x and z are the coordinates of the position of the pivot, I A = Ī + md 2 is the mass moment of inertia of the body about its pivot A. Consider the high-frequency, high-amplitude input force and couple in the form where ϕ 1 (t) and ϕ 2 (t) are zero-mean, T-periodic functions and F 0 ≥ 0 and M 0 ≥ 0 are constants.Following 17,47 , the averaged dynamics of the system is where and where µ 11 ≥ 0 and µ 12 are determined using the periodic input functions ϕ 1 (t) and ϕ 2 (t) [see Eq. ( 33)] 17,47 .
For a brief review of the averaging technique used here, see 10 , Sec.A.2.
Using the averaged dynamics (3), it can be shown that the orientation θe can be an equilibrium of the system if and only if Note that Eq. ( 5) does not guarantee the stability of the equilibrium.Using linearization of the averaged dynamics, the equilibrium θe is stable if the following inequality is satisfied Consider the pendulum without the couple M, that is, M 0 = 0 .For this case the necessary force amplitude to stabilize the pendulum in an orientation θe in its stabilizable set 10 is It is evident that the pendulum with no couple M can be stabilized in any orientation , besides a third region entirely in the upper half-plane.In this section and Section "Pitch dynamics of hovering insects", only the equilibria of the pendulum in the lower half-plane are considered.
(1) www.nature.com/scientificreports/ For the case of a horizontal input force only, i.e., β = 0 and M 0 = 0 , the pendulum can be stabilized in any orientation in the lower half-plane, that is, − π 2 < θe < π 2 using a force amplitude Equation (8) suggests that to stabilize the pendulum in a non-vertical orientation in the lower half-plane using only a horizontal input force, the required force amplitude F 0 must be greater than a minimum force amplitude F m = mg Ī 2µ 11 d .For any force amplitude F 0 ≤ F m the pendulum remains in the downright orientation θ = 0 , on average.Note that if F 0 = 0 , then Eq. ( 5) cannot be satisfied except for θe = 0 or θe = 180 • , that is, the downright and upright orientations.This means it is not possible to stabilize the pendulum in a non-vertical orientation using a zero-mean couple M only.
Using a zero-mean force F and a zero-mean couple M, however, the pendulum can be stabilized in a nonvertical orientation with a smaller or larger force amplitude F 0 (depending on the force and couple zero-mean functions ϕ 1 (t) and ϕ 2 (t) ) compared to the system with no couple.For example, consider the harmonic force and couple functions ϕ 1 = cos(t) and ϕ 2 = cos(t + ψ) .For these functions, one determines µ 11 = 1 4 and µ 12 = 1 4 cos ψ .Therefore, for example, for the case of a horizontal force ( β = 0 ), using (5), the required force amplitude to stabilize the pendulum in an orientation θe is determined to be It is evident that for ψ = ±90 • the couple M does not have any effect on the required force amplitude F 0 , for ψ = 0 the force amplitude F 0 is maximum, and for ψ = 180 • the force amplitude F 0 is minimum.Therefore, to stabilize the pendulum in an orientation θe using a zero-mean horizontal force and a zero-mean couple, by choosing ψ = 180 • , the task can be accomplished with a smaller force amplitude.
For any value of ψ , in general, the minimum horizontal force amplitude is determined using θe = 0 , and the result is It is noteworthy that, in general, the equilibrium set and the stabilizable set (see 10 ) of the pendulum with both force and couple inputs is different from those of the pendulum with only force input and the sets depend on the physical parameters of the pendulum and inputs.Depending on the couple amplitude M 0 and the periodic functions ϕ 1 (t) and ϕ 2 (t) , the input couple may cause the equilibrium and stabilizable sets of the pendulum to expand or shrink.
As mentioned, using a horizontal force only, the equilibrium and stabilizable sets of the 3-DOF pendulum are the lower half-plane.However, by adding a zero-mean couple, the pendulum can be stabilized in the upper half-plane as well.Figure 2 shows the time history of the pendulum stabilized at the desired orientation θe = 150 • in the upper half-plane, on average, using a zero-mean horizontal force (i.e., β = 0 ) and a zero-mean couple.The physical parameters are m = 0.2 kg , d = 0.2 m , Ī = 0.05 kg.m 2 , c = 0.1 N.s/m , c t = 0.05 N.m.s/rad , ω = 200 rad/s , ϕ 1 (t) = cos t , ϕ 2 (t) = − cos t (and therefore, ψ = 180 • ) , and M 0 = 0.5 N.m .The initial (8) . Time history of the pendulum orientation stabilized at θe = 150 • using a zero-mean horizontal force and a zero-mean couple.Solid-green: original system, solid-red: averaged system, dashed-black: desired orientation.www.nature.com/scientificreports/conditions are x(0) = z(0) = 0 , θ(0) = 160 • , and zero initial velocities.From equation ( 5) one determines two values for the required force amplitude, F 0 = 0.468 N and F 0 = 2.418 N .In the simulations the former is used.Note that without the input couple, the desired orientation θe = 150 • will not belong to the equilibrium set of the system.

Pitch dynamics of hovering insects
This section discusses the effects of the wing inertial forces on the body angle of hovering insects and the dynamics of the pitch motion of hovering insects.The body of a hovering insect can be considered as a 3-DOF pendulum, similar to what was discussed in Section "The 3-DOF force-input pendulum", with the aerodynamic forces and couples and inertial forces due to flapping of the wings acting on it.To discuss the effects of the wing inertial forces, in the first part of this section, all the aerodynamic forces and moments, except the average lift, are neglected and only the inertial forces due to the accelerating wings are considered as the input forces acting on the body.(The authors are aware that neglecting the aerodynamic forces in insect flight may seem surprising and is not justified for insect flight analysis.However, this is to discuss the effects of the wing inertial forces only. Besides, a considerable part of the aerodynamic forces, i.e., the averaged lift, is still considered).In the second part, besides the wing inertial forces, using a simple quasi-steady aerodynamic model, an approximation of the aerodynamic forces and moments are also considered acting on the body and the pitch stability and body angle of five different insect species are determined.

The effect of wing inertial forces on the pitch dynamics
Consider the hovering insect depicted in Fig. 3 with a body mass m, mass moment of inertia Ī of the body about its center of mass G which is located at a distance d from the y-axis passing through the wing joints, and with the midpoint between the two wing joints at A on the y-axis, and therefore AG = d .The pitch angle θ of the body is defined as the angle of the line AG with the vertical.Suppose that the wings, each of mass m w , length R, and average chord length c , perform a harmonic flapping in the form where φ(t) is the flapping (stroke) angle measured from the horizontal y-axis, φ 0 > 0 is the flapping (stroke) amplitude, and ω is the (usually high) flapping frequency.For simplicity, assume that when φ = 0 , the center of mass of each wing is on the y-axis.Therefore, the center of mass C of each wing moves in a plane, called the stroke plane, which passes through the wing hinge and has an angle β with the horizontal (see Fig. 3).The total inertial force acting on the body due to the accelerating wings is determined to be where r c is the distance between the wing joint A and its center of mass C (see Fig. 3), and the zero-mean, 2π -periodic function ϕ(t) is www.nature.com/scientificreports/ The inertial force F of the wings is in the form of (2) with F 0 = 2m w r c φ 0 ω .The force F, which lies in the stroke plane, has a constant angle β with the horizontal.Besides the wing inertial forces, the aerodynamic forces and couple are also applied to the body at point A. During hover, the vertical component of the aerodynamic forces, called lift (L) in this paper, is equal to the total weight of the insect, on average, i.e., L = m t g .Also, with the considered wing kinematics (11), the horizontal component of the aerodynamic forces (the projection of the aerodynamic force in the x-y plane), called drag (D) in this paper, and its x-component D x , are zero on aver- age, i.e., D = Dx = 0 .Since the lift and drag are in the same order of magnitude and during hover L = m t g , the aerodynamic forces during hover are in the order of the weight of the insect.
Neglecting the effects of the aerodynamic forces, except the average lift L , the equations of motion of the insect body are in the form where m t = m + 2m w is the total mass of the insect, I A = Ī + md 2 is the mass moment of inertia of the body about the y-axis passing through the wing hinge A, and is the mean moment due to the weight of the wings (which may be ignored due to being small).Equations ( 14) are in the averaging form presented in 16,17 .Using the averaged dynamics, which are not presented here, the required inertial force amplitude F 0 for stabilizing the insect body in an orientation θe in its stabilizable set of the system is determined to be where the parameter µ > 0 is determined using the periodic function ϕ(t) in ( 13) 47 .For a certain value of the stroke plane angle β , the minimum value of F 0 to stabilize the body in a non-vertical orientation 0 < θ < 90 • + β is where 0 ≤ θ m < 90 • is determined from the equation The morphological properties of five insect species, namely, hawkmoth (HW), hoverfly (HF), dronefly (DF), honeybee (HB), and bumblebee (BB) used in this section are presented in Tables 1 and 2. The data for each of the mentioned insect species are taken from 23,37,[48][49][50] , respectively.The bold data could not be found in literature and are estimated.The values of r p are considered around 60%-70% of the wing length 51 .In Table 3, the total weight of the insect W t = m t g , the real amplitude of the total wing inertial forces determined using (12), that is, F t = F 0 ω = 2m w r c φ 0 ω 2 , the ratio ρ t = F t W t , and the input parameter µ determined using the periodic function ϕ(t) in ( 13) for the five insect species are presented.The ratio ρ 0 = F 0 F m , where F 0 = 2m w r c φ 0 ω and F m is the minimum force amplitude determined using (17), is also presented in Table 3.
There are two obvious conclusions from the values presented in Table 3. First, since during hover the aerodynamic forces are in the order of the body weight, from the values of the ratio ρ t it is concluded that during hover, the amplitude of the wing inertial forces may be larger than the aerodynamic forces and should not be ( 13) ϕ(t) = sin t cos φ(t) + φ 0 cos 2 t sin φ(t).
The morphological properties of the body and flapping frequency of the five insect species.m: body mass, Ī : mass moment of inertia about the center of mass, d: the distance from the center of mass to the y-axis, ω : flapping frequency.neglected in stability analysis of hovering flight.And second, the values of the ratio ρ 0 suggest that since for the five insect species ρ 0 < 1 , the inertial forces, though considerable in magnitude, are not large enough to stabilize the body in a non-vertical orientation.They only provide around 10%-30% of the required vibrational force to put the body in a non-vertical orientation.It must be emphasised that, the results are based on a number of assumptions discussed earlier, such as harmonic flapping of the wings and neglecting the fluid added mass.
From the numerical values in Table 3, it is also evident that, since the aerodynamic lift and drag are in the range of the weight, and therefore smaller than the wing inertial forces, adding the aerodynamic forces only (and not aerodynamic moments) does not have a considerable effect on the results and does not change the result that vibrational forces cannot stabilize the body in a non-vertical orientation.
Considering the results presented in Section "The 3-DOF force-input pendulum" about the role of a zeromean couple on the necessary force amplitude, one may think of the role of the aerodynamic couple on the pitch stability.The flapping kinematics (11) considered in this section, generates a zero-mean aerodynamic moment.Since the aerodynamic moment generated due to the aerodynamic forces is a function of the square of the wing velocity, i.e., φ2 , it does not have a phase difference with the centripetal acceleration, and its phase difference with the tangential acceleration φ is 90 • .Therefore, the phase angle between the inertial forces and aerodynamic moment is in the range of zero and 90 • , and based on the discussions in Section "The 3-DOF force-input pendu- lum", the zero-mean aerodynamic couple does not help reducing the necessary force amplitude for stabilization of the body in a non-vertical orientation.In other words, with the symmetric flapping kinematics (11), all the aerodynamic and inertial forces and moments together are not large enough to stabilize the body of a hovering insect in a non-vertical orientation.The results suggest that though in the real world the bodies of hovering insects are stabilized in a non-vertical orientation, they are not vibrationally stabilized.After introducing the aerodynamic parameters in Section "Hovering insects dynamics with aerodynamic and inertial forces", the results of a numerical simulation will be presented which confirm this claim (see Fig. 7).

Hovering insects dynamics with aerodynamic and inertial forces
In this section, using a simple quasi-steady aerodynamic model and considering the wing inertial forces acting on a hovering insect, the averaged pitch dynamics are derived and the body angle during hover is predicted.The symmetric flapping kinematics (11) is not the real kinematics that insects perform during hover.A more realistic kinematics is in the asymmetric form 43 where φ > 0 is a constant, called the mean stroke angle, φ 0 > 0 is the stroke amplitude, and ζ(t) is a zero-mean, T-periodic function.In this section the harmonic function ζ(t) = sin t is used which is an acceptable estimation of flapping kinematics for most of the insects.Using the asymmetric kinematics (19), insects generate a nonzeromean aerodynamic couple which opposes the moment of their weight about the wing hinges and stabilizes their body in a non-vertical orientation 43 .Compared to (14), a more general dynamics of the insect body presented in Fig. 3 is The morphological properties of the wings and flapping kinematics parameters of the five insect species.m w : mass of one wing, R: wing length, c : mean chord length, r c : distance from the wing hinge to wing center of mass, r p : distance from wing hinge to center of pressure in y-direction, c p distance from center of pressure to the y-axis, φ 0 : stroke amplitude, φ : mean stroke angle, β : angle of stroke plane.Estimated values are in bold.where F x and F z are the total (aerodynamic and inertial) forces in the x-and z-directions, M y is the total aero- dynamic moment acting on the body about the y-axis, and Mw is defined in (15).Assuming a quasi-steady aerodynamic model, the aerodynamic forces are proportional to the square of the wing velocities, φ2 .Therefore, the lift force L is where L 0 is a constant.The average lift L is determined being Since during hover L = m t g , one determines L 0 = 2m t g φ 2 0 ω 2 , and the lift force can be written in the form During hover the drag D is a zero-mean force which is also proportional to φ2 .For many insects, the lift-to- drag amplitude ratio during hover is almost one.Therefore, this paper assumes that the amplitude of the drag is equal to the amplitude of the lift, that is, where sgn(•) is the signum function.Therefore, the x component of the drag is To determine the aerodynamic moment, suppose that the pressure center of the wing is located at point P, a distance r p and c p from the wing root, as shown in Fig. 3. Since the lift and drag are assumed equal, the total aerodynamic moment acting on the body about the y-axis is determined to be where α = α 0 sgn( φ) = α 0 sgn(cos ωt) is the wing pitch angle measured from the vertical.For the five insect species considered in this paper, it is assumed that α 0 = 45 • .Adding the wing inertial forces F from ( 12), the forces F x and F z are It is evident that F x is zero-mean, however, F z and M y are not.To transform the equations of motion (20) into an appropriate form for averaging, the total forces and moment can be rewritten in the form where using T = 2π ω , and where the zero-mean, 2π-periodic functions ϕ i (t) , i ∈ {1, 2, 3} , are Replacing the total forces and moment from ( 28) into (20), the equations of motion of the insect body are (20) M y =2m t g c p sin α cos φ(ωt) + r p sin φ(ωt) cos β + sin β cos φ(ωt) sgn(cos ωt) − c p cos α sin β − cos β cos φ(ωt) sgn(cos ωt) cos 2 ωt (27) www.nature.com/scientificreports/Equations ( 31) are in the averaging form presented in 16,17,47 .The determined averaged dynamics are where and and where, following 47 , the parametersµ ij , i, j = 1, 2, 3 , are determined using the functions ϕ i (t) , i = 1, 2, 3 , in the form Using the averaged dynamics (32), the equilibrium orientation of the body, on average, is determined from equation θ = 0 when replacing the average velocities with zero, i.e., ẋ = ż = 0 and θ = 0 , which is in the form One may also use the approximation m t ≈ m to write the averaged dynamics (32) and the equilibrium deter- mining Eq. ( 34) in slightly simpler forms.
Using the state vector ȳ = ( θ , ẋ, ż, θ) T , the averaged dynamics (32) can be written as a first order system, which then can be linearized about the equilibrium point ȳe = ( θe , 0, 0, 0) T , where θe is the equilibrium orienta- tion determined from (34).The linearized averaged dynamics shows that for each of the five insect species with their morphological properties presented in Tables 1 and 2, the determined equilibrium is stable.Therefore the original system possesses a stable periodic orbit in a small neighborhood of that equilibrium.The damping coefficients and the eigenvalues of the state matrix of the linearized averaged dynamics for the five insect species are presented in Table 4.The eigenvalues suggest that, using the simple aerodynamic model, the pitch motion of those species is open-loop stable.However, as mentioned in Section "Introduction", high-fidelity aerodynamic models suggest that hovering flight may be open-loop unstable due to aerodynamic disturbances such as wind 32,35 .
Figure 4 shows the asymmetric flapping (19), the nonzero-mean forces during hover, i.e., lift and weight, and the equivalent average force-couple system acting on the body of a hovering insect due to asymmetric flapping during one period.As shown in that figure, the lift may be replaced by an equivalent force-couple system, on average, with a force L and couple M.
Figures 5 and 6 show the total moment M y and forces F x and F z and the simulation results for hovering of the hawkmoth with its morphological parameters presented in Tables 1, 2, and 4. The initial conditions of the simulations presented in Fig. 6 are x(0) = z(0) = 0 , θ(0) = 30 • , and zero initial velocities.The stability of the equilibrium point of the averaged dynamics and the corresponding periodic orbit of the original time-periodic system can be seen in Fig. 6.
As mentioned in Section "The effect of wing inertial forces on the pitch dynamics", the results suggest that without the flapping asymmetry, the aerodynamic and inertial forces are not enough to put the body of a hovering insect in a non-vertical orientation, as seen in real world.To show this, the results of numerical simulation of the dynamics of a hovering hawkmoth with symmetric flapping kinematics, that is, φ = 0 , is presented in Fig. 7.The parameters and initial conditions are the same as used to generate the results in Fig. 6, except that the mean stroke angle is considered to be zero ( φ = 0 ).It is evident that the body cannot be stabilized anymore and moves to an almost downright orientation.
Table 4 also presents the average equilibrium orientation θe of the body during hover determined using (34) for the five insect species.The bold data could not be found in the literature and are estimated.The orientation angle θ considered in this paper is the angle of the line AG, and not the body itself, with the vertical (see Fig. 3).This angle is smaller than the real body angle.As an estimation of the body angle determined using the averaged dynamics, one may use the geometry shown in Fig. 8 where the insect body is shown as an ellipse.To determine the body angle of each insect species, one may use the determined equilibrium orientation θe and the two lengths d and a shown in Fig. 8, and determine the body angle χ det using (31)   m t ẍ + md θ cos θ =md θ2 sin θ − c ẋ + ωϕ 1 (ωt)   Top: solid-blue is the moment M y , dashed-red is the average moment M , and dot-dashed-black is mgd for comparison.Bottom: solid-blue is F x , dashed-red is F z , and dot dashed-black is mg for comparison.4 also presents the real body angle χ obs of the five insect species observed during experiments and reported in the literature used for the morphological data and also in 43,52,53 .The determined body angles using the averaged dynamics for the five insect species show good agreement with the observed body angles.The results from the stability analysis of the pitch motion of hovering insects presented in this section show that the main parameter determining the pitch dynamics and and the average body angle of hovering insects is the average moment M generated by asymmetric flapping which counteracts the moment due to the insect weight about the wing hinges.The nonzero average aerodynamic moment M is the result of an asymmetric flap- ping kinematics, such as (19).The wing inertial forces, which as shown, are larger than aerodynamic forces in amplitude, also play a minor role in the equilibrium orientation of the body during hover.

Experimental results
To demonstrate the effect of the mean stroke angle in stability of the pitch dynamics of hovering insects, we designed a flapping wing device consisting of a 1-DOF main body with two wings attached to it, as shown in Fig. 9.The body, which represents the insect body, is free to rotate about a fixed horizontal shaft.Each wing consists of a light flexible membrane attached to a rigid arm (frame).The wings are driven back and forth by a DC motor and a Scotch yoke mechanism 54 .The flapping mechanism can be adjusted to flap with either a zero    4. The damping coefficients c (N s/m) and c t (N m s/rad) , the eigenvalues i of the linearized averaged dynamics, the length a, the equilibrium orientation θe , the determined body angle χ det , and the observed body angle χ obs of the five insect species.Estimated values are in bold.or nonzero mean stroke angle, that is, symmetric or asymmetric flapping kinematics.Using carefully selected counterweights, the system is made slightly heavier on one side.Therefore, the center of mass of the device is located outside of its axis of rotation, that is, the fixed horizontal shaft.Without flapping, the system is a 1-DOF pendulum that is stable in its vertical orientation.The goal of the experiments is to show that the body can be stabilized in a non-vertical orientation while flapping with a nonzero mean stroke angle (asymmetric flapping).However, it cannot be stabilized using flapping with a zero mean stroke angle (symmetric flapping).In other words, the goal is to experimentally show that the aerodynamic moment generated by asymmetric flapping can stabilize the pitch dynamics of the main body in a non-vertical orientation.
As shown in the video accompanying this paper, in the first experiment we used a symmetric flapping kinematics and tried to stabilize the body in a non-vertical orientation.However, symmetric flapping was not able to stabilize the pitch dynamics and the body remained close to the vertical orientation on average.
In the second experiment we used an asymmetric flapping kinematics that was able to stabilize the pitch dynamics in a non-vertical orientation.Using a higher flapping frequency, we were to stabilize the body in an almost horizontal orientation.To show that the pitch stability is caused by the aerodynamic moment generated by asymmetric flapping, and not the vibrational effects of the wing inertial forces, in the third experiment the membranes are removed from the wing arm (frame).The membranes are light and their total mass and inertial forces are negligible compared to the mass and inertial forces of the wing frame and other reciprocal parts.The membranes generate almost the entire aerodynamic forces during flapping, and by removing them the aerodynamic effects vanish.As is evident in the video of the third experiment, though we used the same asymmetric flapping kinematics as in the second experiment, the pitch dynamics cannot be stabilized in a non-vertical orientation without the aerodynamic moment.Although in the experimental device the main body is a 1-DOF pendulum, compared with the 3-DOF body of a hovering insect, the experiments clearly show that the asymmetric flapping kinematics plays the most important role on the stability of the pitch dynamics of hovering insects.

Conclusions
To determine the effects of wing inertial forces and flapping kinematics on the pitch dynamics of hovering insects, the body of a hovering insect is considered as a 3-DOF pendulum with inputs consisting of aerodynamic forces and moments and the vibrational wing inertial forces.Using numerical values, it was shown that, in general, the inertial forces due to the flapping of the wings are larger than the aerodynamic forces during hovering flight.However, the inertial and aerodynamic forces and moments together may not be large enough to vibrationally stabilize the insect body in a non-vertical orientation.Instead, the pitch dynamics of insect bodies during hover are mainly governed by two counteracting moments, the moment of the body weight about the wing hinges and the nonzero-mean aerodynamic moment generated due to an asymmetric flapping kinematics.Using a simple aerodynamic model and numerical values for five insect species, it was shown that their average body angle during hover predicted by the analysis presented in this paper agrees with the body angles of real hovering insects observed in experiments.The results of this paper suggest that the two main parameters determining the body

Figure 3 .
Figure 3.The insect parameters and the inertial force F. The aerodynamic forces and moments, except the average lift L , are not shown.The points C and P are the center of mass and center of pressure of the wings, respectively.

Figure 4 .
Figure 4. (a) Top view of the asymmetric flapping (19), (b) the nonzero-mean forces acting on the body of a hovering insect with the asymmetric flapping kinematics (19) during one period, and (c) the equivalent average force-couple system of the lift.

Figure 5 .
Figure 5.Total moment and forces acting on the body of a hovering hawkmoth over one flapping period.Top: solid-blue is the moment M y , dashed-red is the average moment M , and dot-dashed-black is mgd for comparison.Bottom: solid-blue is F x , dashed-red is F z , and dot dashed-black is mg for comparison.

Figure 8 .
Figure 8.The determined body angle χ det and the equilibrium orientation θe .

Table 3 .
The weight and inertial forces of the five insect species.